Electroweak Baryogenesis Mechanism

• Electroweak baryogenesis is a mechanism that generates the Universe’s baryon asymmetry during a strong first-order phase transition driven by beyond-the-Standard-Model modifications to the Higgs potential.
• CP-violating interactions at the bubble walls create chiral imbalances that, via nonperturbative sphaleron transitions, convert into a net baryon number.
• Observable signatures include altered Higgs couplings at colliders, sensitive electric dipole moment experiments, and potential gravitational wave signals.

Electroweak baryogenesis (EWBG) is a dynamical mechanism for generating the baryon asymmetry of the Universe (BAU) during the electroweak phase transition (EWPT). The core concept is that new physics near the electroweak scale provides both additional sources of CP violation and modifies the Higgs potential to render the phase transition strongly first order. As the expanding phase boundaries (bubble walls) sweep through the primordial plasma, CP-violating interactions produce chiral charge asymmetries that are subsequently converted into net baryon number by nonperturbative sphaleron transitions. This scenario is testable by collider measurements, electric dipole moment (EDM) searches, and (in some cases) gravitational wave observations. Implementing EWBG, however, faces both theoretical and experimental challenges, particularly in quantifying CP-violating sources and reconciling them with strong EDM bounds.

1. Sakharov Conditions and Theoretical Framework

EWBG is constructed to fulfill the Sakharov conditions: baryon number violation, C and CP violation, and departure from thermal equilibrium. In the Standard Model, baryon number violation is provided by electroweak sphaleron transitions, but the amount of CP violation (dominated by the CKM phase) and the nature of the EWPT (a crossover for $m_H = 125$ GeV) are insufficient. EWBG thus requires beyond-the-Standard-Model (BSM) extensions that achieve:

• A strongly first-order EWPT, typically driven by new scalars (singlets, doublets, or composite states) that modify the finite-temperature effective Higgs potential.
• New CP-violating phases, often realized via higher-dimensional operators, extended Higgs sectors, or new interactions in the Yukawa or gaugino sectors (Morrissey et al., 2012, Matsedonskyi et al., 2022, Harling et al., 2023).

The finite-temperature effective potential is crucial for EWBG studies: $$V_{\rm eff}(\phi,T) = V_{\rm tree}(\phi) + \Delta V_{\rm 1\,loop}(\phi,T)$$ with temperature-dependent corrections yielding terms such as $-E T\phi^3$ that create a barrier between vacua, supporting a first-order transition (Morrissey et al., 2012, Vis et al., 13 Aug 2025). The requirement that sphaleron washout is suppressed inside bubbles sets a condition $v_n/T_n \gtrsim 1$ for the Higgs VEV $v_n$ at the nucleation temperature $T_n$.

2. Charge Transport, CP Violation, and Quantum Kinetics

EWBG proceeds via CP-violating interactions at the bubble wall, generating an asymmetry in the plasma:

1. Particles scatter off walls with space-time–varying masses (often with spatially varying complex phases), yielding CP-violating force terms.
2. The resulting left-handed chiral charge diffuses into the symmetric phase, biasing sphaleron transitions.
3. Sphalerons energetically favor transitions that convert this chiral imbalance into net baryon number.

Quantum transport equations are essential for quantitatively accurate predictions. Advanced derivations using the closed time path (Schwinger–Keldysh) formalism yield Kadanoff–Baym equations in Wigner space, which, upon gradient expansion, produce semiclassical Boltzmann-type equations (Konstandin, 2013): $$k_z \partial_z n_0^s - \frac{1}{2}\frac{d|m|^2}{dz}\partial_{k_z} n_0^s - \frac{s}{2k_0} \frac{d^2(|m|^2 \theta')}{dz^2} \partial_{k_z} n_0^s = \mathrm{coll.}$$ where the final term acts as the CP-violating semiclassical force.

Flavor oscillations, especially when nearly-degenerate mass eigenstates are present, can lead to resonant enhancement of the generated chiral charge via nonadiabatic evolution of the density matrix (flavor coherence), as established in (Cirigliano et al., 2011). In particular, when the oscillation length $L_{\rm osc} \sim L_w$ (wall thickness), the full matrix structure of the quantum Boltzmann equations must be retained to capture the feeding of off-diagonal (coherence) components into diagonal charge densities.

A comprehensive calculation involves solving coupled transport equations for the relevant density matrices with consistent power counting in gradient expansion and retaining terms crucial in the resonant regime.

3. Extensions and Novel EWBG Mechanisms

A variety of model extensions and EWBG scenarios are developed:

• Singlet-extended and multi-Higgs models: These supply new tree-level or loop-induced barriers in the thermal potential to support a first-order transition and offer additional CP-violating phases (Morrissey et al., 2012, Damgaard et al., 2015, Curtin et al., 2014, Vis et al., 13 Aug 2025).
• Supersonic and detonation wall regimes: Contrary to the standard EWBG scenario (with subsonic walls), EWBG can proceed with supersonic expansion via nucleation of "symmetric phase" bubbles behind the wall due to plasma reheating. The net baryon asymmetry is determined by the fraction of plasma crossing these symmetric bubbles ("filling factor") (Caprini et al., 2011).
• Two-stage transitions and exotic symmetry breaking: The Universe can undergo sequential transitions, with an initial exotic phase (from a non-Higgs scalar) generating baryon asymmetry, before transitioning to the usual Higgs phase (Blinov et al., 2015).
• Primordial black hole catalysis: Hawking radiation from small black holes creates local regions of restored EW symmetry, enabling baryogenesis via domain walls without the need for thermal first-order transition. Enhanced accretion and lifetime (e.g., in Randall–Sundrum or Brans–Dicke gravity) can yield acceptable baryon–to–entropy ratios for very small CP-violating phases (Aliferis et al., 2014, Aliferis et al., 2020).
• Topological and dark sector mechanisms: Baryogenesis may occur via monopole-induced $B$-violation through Born-Infeld extended gauge sectors (Arunasalam et al., 2018), or with CP violation realized entirely in a dark sector and communicated via new gauge bosons (e.g., leptophilic $Z'$) that induce effective chemical potentials for sphaleron processes (Carena et al., 2018).
• Composite Higgs and high-temperature symmetry non-restoration: The scalar potential may be engineered (e.g., with new fermion species) such that EW symmetry remains broken at high temperatures, allowing baryogenesis at scales well above the usual $\sim 100\,\text{GeV}$ and relaxing constraints from dilaton searches (Harling et al., 2023, Matsedonskyi et al., 2022).
• Domain wall–mediated baryogenesis: Both embedded domain walls stabilized by plasma effects and minimal domain walls from spontaneous breaking of a $\mathbb{Z}_2$-symmetric singlet can provide EW-symmetric cores for unsuppressed sphaleron processes, with wall dynamics, wall width, and CP-violating operator structure dictating the baryon yield (Schröder et al., 19 Apr 2024, Azzola et al., 13 Dec 2024).

4. Experimental Probes and Phenomenological Signatures

EWBG offers multiple avenues for experimental investigation:

• Collider signatures: Extended scalar sectors lead to modified Higgs couplings (to gauge bosons, fermions, and self-couplings), new scalar resonances, and possible exotic Higgs decays (such as $h \to SS$). Direct vector boson fusion (VBF) production of scalars with missing energy and precision measurements of processes like $Zh$ associated production can probe scenarios even in "nightmare" cases with suppressed direct signals (Curtin et al., 2014, Xie, 2020).
• Electric Dipole Moment (EDM) searches: New CP-violating operators generically induce EDMs at one or two loops (notably via Barr–Zee diagrams), providing extreme sensitivity to CP phases (Bian et al., 2014). Cancellation mechanisms between multiple operator contributions or shifting CP violation to dark sectors may evade the most stringent bounds. Recent advances in transport theory demonstrate that first-order gradient CP-violating sources relax EDM constraints compared to earlier analyses, thereby keeping EWBG viable in light of updated limits (Li et al., 30 Apr 2024).
• Gravitational waves: A strongly first-order EWPT produces a stochastic gravitational wave background with a peak frequency and amplitude set by bubble nucleation rate, wall velocity, and latent heat. LISA and future GW interferometers probe parts of the parameter space, with stronger transitions associated with detectable signals (Vis et al., 13 Aug 2025, Xie, 2020, Harling et al., 2023).
• Low-energy/high-intensity experiments: Fifth force searches (for sub-eV mass mediators), rare meson decay experiments, and lepton flavor universality tests can be sensitive to light singlet sectors relevant for domain wall EWBG (Azzola et al., 13 Dec 2024).
• Astrophysical and cosmological probes: Relic backgrounds, dark matter considerations (in dark CPV scenarios), and modifications to cosmological observables offer additional constraints or diagnostics.

5. Quantitative Approaches and Theoretical Developments

Contemporary EWBG research relies on a set of quantitative tools and best practices:

• Effective Potential Calculation: Inclusion of one- and two-loop corrections, daisy resummation, and treatment of gauge dependence (Vis et al., 13 Aug 2025, Morrissey et al., 2012).
• Bubble Nucleation Analysis: Calculation of bounce solutions and action $S_3(T)/T$ for determining nucleation rates and critical temperatures. Barrier-generating cubic terms in the potential are analyzed for their thermal origin and dependence on new couplings (Curtin et al., 2014).
• Quantum Boltzmann Equations: Use of Schwinger–Keldysh CTP formalism for systematic gradient expansion, correct power counting in length scale ratios (e.g., $L_{\rm osc}/L_w$), and retention of flavor coherence where appropriate (Cirigliano et al., 2011, Konstandin, 2013).
• Transport and Diffusion: Coupled differential equations for chiral and baryon densities, accounting for diffusion constants, washout rates, and sphaleron process rates (often extracted from lattice studies).
• Model Parameter Scans: Direct mapping of viable EWBG parameter space onto collider, EDM, and GW observables, with constraints and projections from experimental searches (Curtin et al., 2014, Xie, 2020, Azzola et al., 13 Dec 2024).

6. Challenges, Limitations, and Future Directions

While EWBG remains an active and promising research direction, several obstacles are significant:

• The simultaneous realization of a strong first-order EWPT and sufficient CP violation without violating EDM bounds is highly nontrivial; most minimal scenarios are pushed into fine-tuned regions by LHC and EDM data (Vis et al., 13 Aug 2025, Bian et al., 2014).
• Gauge dependence in potential and bubble nucleation calculations, breakdowns in gradient expansions (especially in the resonant regime), and uncertainties in the calculation of wall velocity and thickness all introduce theoretical uncertainties.
• In models with heavy new states, validity of effective field theory approaches can be compromised by modest scale separations and large dimensionful couplings (Damgaard et al., 2015).
• Alternative mechanisms, including lepton sector CPV, two-stage transitions, or new topological defects, remain under active exploration for both theoretical viability and distinctive phenomenological signatures (Blinov et al., 2015, Schröder et al., 19 Apr 2024).

Planned and proposed experimental efforts (future hadron colliders, lepton colliders, EDM experiments with improved sensitivity, and GW detectors) will play a decisive role in falsifying or substantiating large regions of EWBG parameter space. Progress in lattice gauge theory, improved quantum kinetic treatments, and more precise modeling of BSM scalar sectors is expected to further sharpen both predictions and constraints.

7. Summary Table: Selected Mechanisms and Experimental Probes

Mechanism/Class	Key Prediction	Major Experimental Probes
Singlet/doublet scalar models	Modified Higgs couplings, scalar resonances	LHC/HL-LHC/100 TeV, precision Higgs
Resonant flavor oscillations	Enhanced CPV in nearly degenerate sectors	EDMs, indirect flavor probes
Supersonic bubble expansion	Gravitational waves, reduced EDM signal	GW detectors, less EDM tension
Two-stage symmetry breaking	Light new scalars, altered phase history	Collider searches, precision EW tests
Primordial black hole scenarios	Baryogenesis without first-order phase transition	Cosmological/astrophysical, indirect EDM
Dark sector CPV	Leptophilic Z', dark matter, suppressed EDMs	Rare decays, DM, axion, lepton flavor
Domain wall baryogenesis	Fifth forces, light scalars, meson decays	Torsion balance, B-meson, GW (domain walls)

The landscape of EWBG now encompasses a wide range of theoretical mechanisms and model implementations, each furnishing distinctive experimental manifestations. The ongoing convergence of theoretical advances in out-of-equilibrium quantum field theory and high-precision experimental programs continues to define the direction and prospects of research in this dynamic field.